Equalization using serial localization with indecision

ABSTRACT

A receiver includes a constellation processing module and a plurality of equalization stages. The constellation processing module groups points of a constellation associated with a transmitted signal into a plurality of subsets. At least two adjacent ones of the subsets have one or more common constellation points so that the at least two adjacent subsets overlap. The constellation processing module also determines a centroid-based value for each of the subsets of constellation points and groups the centroid-based values into one or more sets. Each of the equalization stages except for a the last equalization stage localizes a search for a final symbol decision using the set of centroid-based values input to or selected by the equalization stage as constellation points. The last equalization stage determines the final symbol decision using the subset of constellation points input to or selected by the last equalization stage.

TECHNICAL FIELD

The present invention generally relates to equalization, and moreparticularly relates to equalization based on serial localization withindecision.

BACKGROUND

MLSE (Maximum Likelihood Sequence Estimation) is a demodulationtechnique that also suppresses ISI (Inter-Symbol Interference) from asignal which is modulated in accordance with a particular constellationand transmitted over a channel. ISI causes the equalization complexityto increase as a power of the constellation size. Relatively largesignal constellations such as 16-, 32- and 64-QAM (Quadrature AmplitudeModulation) have been adopted in EDGE (Enhanced Data Rates for GSMEvolution), HSPA (High Speed Packet Access), LTE (Long Term Evolution)and WiMax (Worldwide Interoperability for Microwave Access). In HSPA,multi-code transmission creates even larger effective constellations.Also, MIMO (Multiple-Input, Multiple-Output) schemes with two or morestreams have been adopted in HSPA, LTE and WiMax. MIMO implementationsalso yield relatively large effective constellations. ISI causesequalization complexity to further increase when any of these techniquesoccur in combination, e.g. multi-code and MIMO.

In the ISI context, the ideal equalization scheme is MLSE, in the senseof maximizing the probability of correctly detecting the transmittedsequence of symbols, or sequences of symbols in the MIMO case. However,the complexity of MLSE increases substantially as a function of the sizeof the modulation constellation and/or because of the exponentialeffects of MIMO or multi-codes to the point where MLSE becomesimpractical. Less complex solutions are available such as, DFSE(Decision-Feedback Sequence Estimation), DFE (Decision-FeedbackEqualization), etc. Each of these solutions attempts to strike a balancebetween accuracy and complexity.

For a symbol-spaced channel model with memory M in a MIMO environment,the system model is given by:

r _(k) =H _(M) s _(k-M) + . . . +H ₁ s _(k-1) +H ₀ s _(k) +V _(k)  (1)

Here the element H_(m, i, j) of H_(m) describes the channel fromtransmit antenna j to receive antenna i at a delay of m symbols. Thechannel matrices are assumed to be constant over the duration of a burstof data, which will be equalized in one shot. The signal s_(k) hassymbol constellation Q of size q. The noise v_(k) is white and Gaussian.

The general ISI scenario includes MIMO. Without much loss of generality,consider the case of a single transmitted signal. For a channel withmemory M, MLSE operates on the standard highly regular ISI trellis withq^(M) states, and q^(M+1) branches per stage. The storage complexity ofMLSE is roughly driven by the number of states, and the computationalcomplexity by the number of branches. As either M or q grows large, thecomplexity explodes. Stage k of the trellis describes the progressionfrom state (ŝ_(k-M) . . . ŝ_(k-1)) to state (ŝ_(k-M+1) . . . ŝ_(k)). Thebranch from (ŝ_(k-M) . . . ŝ_(k-1)) to (ŝ_(k-M+1) . . . ŝ_(k))represents the symbol ŝ_(k). Note that for the ISI trellis, all branchesending in (ŝ_(k-M+1) . . . ŝ_(k)) share the same symbol. For notationalsimplicity, the states at each stage are labeled 0 to q^(M)−1. Eachindex represents a distinct value of (ŝ_(k-M+1) . . . ŝ_(k)). A branchis labeled by its starting and ending state pair (j′,j). For each statej, the fan-in I(j) and the fan-out O(j) are the set of incoming andoutgoing branches, respectively. For the ISI trellis, all fan-in andfan-out sets have the same size q.

The branch metric of a branch (j′,j) at step k in the MLSE trellis isgiven by:

e _(k)(j′,j)=|r _(k) −H _(M) ŝ _(k-M) + . . . +H ₀ ŝ _(k)|²  (2)

Without much loss of generality, the trellis is assumed to start at time0 in state 0. The state metric computation proceeds forward from there.At time k, the state, or cumulative, metric E_(k)(j) of state j is givenin terms of the state metrics at time k−1, and the branch metrics attime k is given by:

$\begin{matrix}{{E_{k}(j)} = {\min\limits_{j^{\prime} \in {I{(j)}}}\left( {{E_{k - 1}\left( j^{\prime} \right)} + {e_{k}\left( {j^{\prime},j} \right)}} \right)}} & (3)\end{matrix}$

In addition, the state in I(j) that achieves the minimum is theso-called predecessor of state j, and denoted π_(k-1)(j). Also, theoldest symbol ŝ_(k-M) in the corresponding M-tuple (ŝ_(k-M), . . . ,ŝ_(k-1)) is the tentative symbol decision looking back from state j attime k. It is possible to trace back a sequence over the differentstates to time 0, by following the chain π_(k-1)(j),π_(k-2)(π_(k-1)(j)), etc. The corresponding symbols ŝ_(k-M), ŝ_(k-M−1),etc, are the tentative decisions of MLSE looking back from state j attime k. In general, looking back from different states at time k, thedecisions tend to agree more the older the symbols. That is, the longerthe delay for a decision, the better. Typically, there is a chosen delayD, and the final decision about symbol ŝ_(k-M-D) is made by tracing backfrom the state (ŝ_(k-M+1) . . . ŝ_(k)) with the smallest state metric.We note again, however, that MLSE has exploding complexity, whether dueto the size of the modulation itself, or to the exponential effect ofISI.

Another conventional equalization technique is MSA (Multi-StageArbitration). MSA involves sifting through a large set of candidates inmultiple stages, where each stage rejects some candidates until a singlecandidate is left after the final stage. One specific example of MSA isgeneralized MLSE arbitration where the first stage is a linearequalizer. The second stage implements MLSE based on a sparse irregulartrellis over a reduced state space. Iterative Tree Search (ITS) has alsobeen used for performing equalization in MIMO QAM environments. ITSexploits the triangular factorization of the channel. In addition, ITSuses the M-algorithm for reducing the search for the best candidate. ITSbreaks down the search further, by dividing the QAM constellation in itsfour quadrants, and representing each quadrant by its centroid inintermediate computations. The selected quadrant itself is subdividedagain into its 4 quadrants, and so on. This results in a quaternary treesearch. Other conventional approaches give particular attention to theadditional error introduced by the use of centroids instead of truesymbols. The error is modeled as Gaussian noise whose variance isdetermined and incorporated in likelihood computations. However, a tightconnection is typically made between the centroid representation and thebit mapping from bits to symbols. That is, if a so-called multi-levelbit mapping is employed, then identifying a quadrant is equivalent tomaking a decision on a certain pair of bits. Such constraints place arestriction on bit mappings, restricting the design of subsets.

SUMMARY

Equalization is performed in a series of stages for suppressing ISI.Each stage attempts to further localize the search for a solution forthe benefit of the next stage, based on input from the previous stage.The equalization structure is generally referred to herein as seriallocalization with indecision (SLI). SLI is a lower complexityalternative to MLSE in the ISI scenario. Viewed in isolation, a givenSLI equalization stage can be quite indecisive, but makes progress andavoids an irreversible wrong decision. A given equalization stagelocalizes the solution by inputting a subset representative of theconstellation and outputting a further reduced subset. Each stage makesa choice among candidate reduced subsets. Indecision arises fromrepresenting the modulation constellation with overlapping subsets.Indecision is beneficial in a multi-stage structure, because indecisiondiscourages an irreversible bad decision in an early stage.

According to an embodiment of a method for equalizing inter-symbolinterference (ISI) in a received signal corresponding to a transmittedsignal carried over a channel, the method includes grouping points of aconstellation associated with a transmitted signal into a plurality ofsubsets. At least two adjacent ones of the subsets have one or morecommon constellation points so that the at least two adjacent subsetsoverlap. A centroid-based value is determined for each of the subsets ofconstellation points and the centroid-based values are grouped into oneor more sets for input to an equalizer having a plurality of stage. Anequalizer is used to equalize the ISI. Each stage of the equalizerexcept for the last stage localizes the search for a final symboldecision using the set of centroid-based values input to or selected bythe stage as constellation points. The last equalization stagedetermines the final symbol decision using the subset of constellationpoints input to or selected by the last stage.

Of course, the present invention is not limited to the above featuresand advantages. Those skilled in the art will recognize additionalfeatures and advantages upon reading the following detailed description,and upon viewing the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a block diagram of an embodiment of a receiverincluding a multi-stage SLI equalizer and a constellation processingmodule.

FIG. 2 illustrates a block diagram of an embodiment of a two-stage SLIequalizer.

FIG. 3 illustrates a diagram of an embodiment of overlappingconstellation subsets for use by a multi-stage SLI equalizer.

FIG. 4 illustrates a diagram of an embodiment of overlapping ASKconstellation subsets for use by a multi-stage SLI equalizer.

FIG. 5 illustrates a block diagram of another embodiment of a two-stageSLI equalizer.

FIG. 6 illustrates a block diagram of an embodiment of an i-th stage ofa multi-stage SLI equalizer including an MLSE component.

FIG. 7 illustrates a block diagram of an embodiment of an N-stage SLIequalizer.

FIG. 8 illustrates a diagram of an embodiment of overlapping QAMconstellation subsets for use by a multi-stage SLI equalizer.

FIG. 9 illustrates a diagram of another embodiment of overlapping QAMconstellation subsets for use by a multi-stage SLI equalizer.

FIG. 10 illustrates a block diagram of an embodiment of an N-stage SLIequalization structure including a pre-filter.

FIG. 11 illustrates a block diagram of an embodiment of an i-th stage ofa multi-stage SLI equalizer including a DFSE component.

DETAILED DESCRIPTION

FIG. 1 illustrates an embodiment of a wireless transmitter 100 incommunication with a wireless receiver 110 over a channel 120. Thereceiver includes a baseband processor 130 and a constellationprocessing module 140 and a multi-stage SLI equalizer 150 included in orassociated with the baseband processor 130. The constellation processingmodule 140 groups points of a constellation associated with atransmitted signal into a plurality of subsets, e.g., subsets of ASKconstellation points, QAM constellation points, etc. At least twoadjacent subsets have one or more common constellation points to ensureoverlap. In some embodiments, all adjacent subsets have one or morecommon constellation points to ensure that all adjacent subsets overlap.In each case, the constellation processing module 140 also determines acentroid-based value for each of the subsets of constellation points andgroups the centroid-based values into one or more sets. The valuesincluded in each set are centroid-based in that they may be actualcentroids, approximations of centroids such as integer values or valuesquantized to a certain finite precision, the closest constellation pointto a centroid, etc. More generally, each subset is assigned arepresentative, which we call a centroid from here on.

The multi-stage SLI equalizer 150 includes a plurality of stages 152,154 for suppressing ISI and demodulating a received signal. Each of theequalization stages 152 except for the last stage 154 localizes thesearch for a final sequence decision using the set of centroid-basedvalues input to or selected by the stage 152 as constellation points.The last equalization stage 154 finalizes the decision using a subset ofthe initial constellation points. This way, each of the equalizationstages 152 except for the last stage 154 further localizes the searchfor a solution using a set of the centroid-based values as constellationpoints, reducing the overall complexity of the equalizer. The last stage154 outputs the final solution based on a subset of the actualconstellation. The constellation processing module 140 ensures that atleast two adjacent subsets of constellation points overlap to reduce thelikelihood of demodulation errors, particularly for the earlierequalization stages as will be described in more detail later herein.

FIG. 2 illustrates an embodiment of a 2-stage SLI equalization structure200 included in the receiver 110 of FIG. 1 for suppressing ISI anddemodulating a received signal r_(k). The received signal r_(k) iscarried over the channel 120 and originally modulated at the transmitter100 using symbol constellation Q. The received signal r_(k) has ISI andis given by equation (1) above. One skilled in the art can readilyexpand the signal model represented by equation (1) to other scenariossuch as MIMO and multi-code transmission. The originally transmittedsignal has symbol constellation Q of size q. The constellationprocessing module 140 of the receiver 110 groups the points ofconstellation Q into a plurality of subsets in a way that ensures atleast two adjacent subsets overlap. The constellation processing module140 also determines a centroid-based value for each of the subsets ofconstellation points and generates an alternative constellation Q′including the centroid-based values, not necessarily belonging to Q, forinput to a first stage 210 of the 2-stage SLI equalization structure200.

The first stage 210 of the SLI structure 200 demodulates the receivedsignal r_(k) and suppresses ISI using the alternative constellation Q′.That is, the first equalization stage 210 uses the centroid-based valuesincluded in Q′ as constellation points to perform sequence estimation.Each point in Q′ represents a subset of clustered points in Q. In oneembodiment, each centroid-based value included in Q′ is the actualcentroid for the points of a particular subset of Q. In anotherembodiment, the centroids are approximated as integer values. In yetanother embodiment, each centroid-based value included in Q′ is theconstellation point of Q located closest to the corresponding centroidvalue. Still other types of values may be used which are derived basedon the centroids determined from the different subsets of Q.

The first equalization stage 210 outputs a symbol decision s_(k)′^([1]),which belongs to Q′. The second equalization stage 220 acceptss_(k)′^([1]) and uses s_(k)′^([1]) to choose a localized subset Q″ of Qas its own constellation. The decision s_(k)′^([1]) output by the firstequalization stage 210 can be interpreted to be the representative of Q″in the first equalization stage 210. The second equalization stage 220outputs the final symbol decision ŝ_(k), which belongs to Q″. The finalsymbol decision ŝ_(k) output by the second stage 220 is determined basedon the original received signal r_(k) and subset Q″, which is selectedbased on the localized symbol decision s_(k)′^([1]) output by the firststage 210. In one embodiment, both equalization stages 210, 220implement MLSE over their respective alphabets, e.g. as previouslydescribed herein. Alternatively, the equalization stages 210, 220implement other types of equalization schemes such as DFSE, DFE,M-algorithm, tree searching, etc.

In one embodiment, the first equalization stage 210 outputs the completesequence of symbol decisions s_(k)′^([1]) in one block, before thesecond stage 220 begins its operation. In another embodiment, the firstequalization stage 210 outputs its symbol decisions s_(k)′^([1])sequentially, based on a decision delay D, as discussed earlier. Thenthe second stage 220 can begin its operation as it sequentially acceptsthe symbols from the first stage 210.

The performance of the 2-stage SLI structure 200 of FIG. 2 is primarilylimited by that of the first equalization stage 210, which in turn isdetermined by the choice of subsets. Performance suffers when noadjacent subsets overlap. Consider the case of disjoint subsets. ML(Maximum Likelihood) receivers implicitly define a decision region(Voronoi region) around each constellation point, consisting of receivedvalues closest to that point than any other. The decision regionboundaries are polyhedrons (made up of sections of hyperplanes). Twoconstellation points x and y are neighbors if their decision regionstouch. The common part is a section of the hyperplane P(x,y) thatbisects the space according to x and y. In degenerate cases, the commonpart can become a line or a point. Now consider two adjacent subsets Xand Y of Q which are available to the first equalization stage 210 ofthe two-stage SLI structure 200. Subsets X and Y have centroids c(X) andc(Y), respectively. Consider neighbor pair (x,y), where x belongs to Xand y belongs to Y. Suppose x is transmitted, and the first equalizationstage 210 makes an error and chooses subset Y instead of subset X. Theeffective decision boundary of the first equalization stage 210 is thehyperplane P(c(X),c(Y)). In contrast, ML would make the choice based onP(x,y). For the sake of comparison, ML can be thought of as making aneffective decision between X and Y. Then the effective decision boundaryis made up of the sections P(x,y) of different nearest neighbor pairs(x,y).

FIG. 3 illustrates the effective decision boundary between adjacentsubsets X and Y in two dimensions, where the hyperplane becomes astraight line and each constellation point is represented by a circle.In contrast, the decision boundary for ML is a piecewise straight jaggedline. The discrepancy between these hypothetical decision boundariesleads to a performance loss in SLI.

Ensuring at least two adjacent subsets have overlap smoothes thedecision boundary discrepancy. In particular, in the two stage SLI,including nearest neighbor symbols pairs in the overlap of adjacentsubsets of the first demodulation stage means that the firstdemodulation stage does not have to make a decision about those symbols.That decision will be made in the second stage.

With SLI, the search is further localized from one stage to the next,but the final decision is not made until the last stage. In particular,by making nearest neighbor symbols belong to multiple subsets, a laterequalization stage (e.g. second stage 220 in FIG. 2) may recover from anerror in an earlier stage (e.g. first stage 210 in FIG. 2). In thiscontext, indecision is beneficial.

FIG. 4 illustrates an exemplary embodiment of an 8-ASK constellationgrouped into three subsets. The 8-ASK constellation is given by:

Q={−7,−5,−3,−1,+1,+3,+5,+7}  (4)

The three overlapping subsets shown in FIG. 4 have centroids given by:

Q′={−4,0,±4}  (5)

The overlap means that the second equalization stage 220 of the SLIstructure 200 of FIG. 2 can often recover from a bad decision by thefirst equalization stage 210. The two outer subsets shown in FIG. 4 areoffsets of one another, and the offset is equal to the centroiddifference. SLI complexity can be further reduced by accounting for thehighly structured nature of these subsets. Of course, less structuredsubsets can also be used with SLI.

FIG. 5 illustrates another embodiment of a 2-stage SLI equalizationstructure 500 for suppressing ISI and demodulating the received signalr_(k). The 2-stage SLI structure 500 shown in FIG. 5 is similar to theone shown in FIG. 2, except it is tailored to the highly structurednature of the ASK constellation subsets shown in FIG. 4. Theconstellation input to or selected by the first equalization stage 510is the set of centroids denoted Q′^([1]). The first equalization stage510 outputs a decision ŝ_(k)′^([1]) which is the centroid-based valueincluded in Q′^([1]) that most closely corresponds to the signal r_(k).The first equalization stage 510 also generates a re-modulated signal{circumflex over (r)}_(k)′^([1]) as a function of ŝ_(k)′^([1]) and thechannel 120 over which the signal is carried as given by:

{circumflex over (r)} _(k)′^([1]) =H _(M) ŝ _(k-M)′^([1]) + . . . +H ₀ ŝ_(k)′^([1])  (6)

where H₀ represents the channel 120 with ISI.

The first equalization stage 510 removes the re-modulated signal{circumflex over (r)}_(k)′^([1]) from r_(k) to generate a modifiedsignal r_(k) ^([1]) for input to the second equalization stage 520 asgiven by:

r _(k) ^([1]) =r _(k) −{circumflex over (r)} _(k)′^([1])  (7)

The modified signal r_(k) ^([1]) is then fed to the second stage 520instead of the original signal r_(k). The second equalization stage 520determines the final symbol decision ŝ_(k) by performing sequenceestimation on the signal r_(k) ^([1]) output by the first stage 510using the subset Q′^([2]) of constellation points input to or selectedby the last stage 520, generating a localized symbol decisionŝ_(k)′^([2]) associated with the second stage 520. A summer 530 includedin or associated with the second equalization stage 520 sumsŝ_(k)′^([1]) and ŝ_(k)′^([2]) to generate the final symbol decisionŝ_(k) as given by:

ŝ _(k) =ŝ _(k)′^([1]) +ŝ _(k)′^([2])  (8)

To account for change to the input of the second equalization stage 520,constellation Q′^([2]) is the subset of Q centered so its centroid isequal to 0. With regard to the subset embodiment shown in FIG. 4,Q′^([2]) is the middle subset. The decision s_(k)′^([2]) output by thesecond equalization stage 520 is an element of Q′^([2]). Consideringagain an exemplary 8-ASK constellation embodiment, Q′^([1])={−4,0,+4}and Q′^([2])={−3, −1, +1, +3}. The second subset Q′^([2]) corresponds tothe 4-ASK constellation. The 2-stage SLI embodiments described hereincan be readily extended to any number of desired stages.

In this case as well, the first stage may output the complete sequenceof symbol decisions in one block, the second stage begins its operation.In another embodiment, the first demodulation stage outputs its symboldecisions sequentially, based on a decision delay D, as discussedearlier. Then the second stage can begin its operation as itsequentially accepts the symbols from the first stage.

FIG. 6 illustrates an embodiment of the i-th stage 600 of an N-stage SLIequalizer structure where i<N. In one embodiment, the i-th stage 600includes an MLSE component 610 which operates over a constellationQ′^([i]) to suppress ISI. Let q′^([i]) denote the size of Q′^([i]). TheISI trellis maintained by the MLSE component derives from Q′^([i]).Particularly, the ISI trellis has (q′^([i]))^(M) states, and(q′^([i]))^(M+1) branches per stage. Each state has fan-in and fan-outsize q′^([i]). The input to the i-th equalization stage 600 is themodified received signal r_(k) ^([i-1]) output by the immediatelypreceding stage (not shown in FIG. 6). The constellation Q′^([i]) inputto the i-th stage 600 includes a set of centroid-based values determinedas previously described herein. The MLSE component 610 outputs symbolsŝ_(k)′^([i]) based on Q′^([i]) and r_(k) ^([i-1]). A re-modulatorcomponent 620 of the i-th stage 600 generates a re-modulated signalgiven by:

{circumflex over (r)} _(k)′^([i]) =H _(M) ŝ _(k-M)′^([i]) + . . . +H ₀ ŝ_(k)′^([i])  (9)

A signal subtractor component 630 of the i-th stage 600 subtracts{circumflex over (r)}_(k)′^([i]) from r_(k) ^([i-1]) to yield a modifiedreceived signal r_(k) ^([i]), which is fed to the next equalizationstage (not shown in FIG. 6).

FIG. 7 illustrates an embodiment of an N-stage SLI equalizationstructure 700. The input to the first stage 710 is the original receivedsignal r_(k) ^([0])=r_(k). For the last stage 730, the constellationQ′^([N]) is a subset of Q. There is no need for a re-modulation block inthe last stage 730. A summer component 740 included in or associatedwith the last equalization stage 730 determines the overall final symboldecision by adding all intermediate symbol decisions as given by:

ŝ _(k) =ŝ _(k)′^([1]) + . . . +ŝ _(k)′^([N])  (10)

In one embodiment, each intermediary stage 720 of the N-stage SLIequalizer 700 has the same structure as the i-th equalization stage 600shown in FIG. 6. According to this embodiment, the i-th intermediarystage 720 localizes the search for the final symbol decision ŝ_(k) bydemodulating a modified version of the received signal r_(k) ^([i-1])output by the immediately preceding stage using the set ofcentroid-based values O′^([i]) input to or selected by the i-thintermediary stage 720 and outputting a localized symbol decisionŝ_(k)′^([i]) as described previously herein.

The i-th intermediary stage 720 also generates a re-modulated signal{circumflex over (r)}_(k)′^([i]) as a function of the channel 120 andthe localized symbol decision generated by the stage 720. There-modulated signal {circumflex over (r)}_(k)′^([i]) is removed from themodified version of the received signal r_(k) ^([i-1]) output by theimmediately preceding stage, e.g. as shown in FIG. 6, to generate anewly modified version of the received signal r_(k) ^([i]) for input tothe stage immediately following the i-th intermediary stage 720.Subtracting {circumflex over (r)}_(k)′^([i]) removes part of thetransmitted signal, which acts as self-interference. This enables laterstages to operate with less self-interference. The constellationQ′^([i]) input to or selected by each of the i intermediary equalizationstages 720 includes centroid-based values, which may or may not belongto Q, whereas the constellation Q′^([N]) input to or selected by thelast equalization stage 730 is a subset of Q.

Broadly, there is no restriction on how the overlapping subsets used forSLI equalization are defined. Subset size can vary, the number ofavailable subsets can change from stage to stage, etc. For the case ofASK, overlapping subsets can be defined in a way that yields a nestedstructure and a three subset representation. Consider the general caseof 2^(L) ASK, having the constellation given by:

Q={−2^(L)+1, . . . ,−1,+1, . . . ,+2^(L)−1}  (11)

Three overlapping subsets are defined, where the first subset containsthe 2^(L-1) negative points. The second includes the 2^(L-1) middlepoints {−2^(L-1)+1, . . . , +^(L-1)−1}, corresponding to 2^(L-1) ASK.The third subset includes the 2^(L-1) positive points. The centroids foreach of the three subsets are −2^(L-1), 0 and +2^(L-1), respectively.The same technique can be used to generate three overlapping subsets for2^(L-1) ASK, and so on. An N-stage SLI equalization structure can bedesigned using these subsets with N≦L. Except for the last stage of theN-stage SLI equalizer, the set of centroids input to or selected by thei-th stage is given by:

Q′ ^([i])={−2^(N-i),0,+2^(N-i)}  (12)

The last stage of the N-stage SLI equalizer has the constellation of2^(L-N+1) ASK. In particular, for N=L−1, Q′^([N])={−3, −1, +1, +3}. Ifthe maximum number of stages N=L is used, then Q′^([N])={−1, +1}. The8-ASK example described above yields a nested subset construction, withL=3 and N=2 stages.

The SLI embodiments described herein can be readily adapted to othermodulation schemes such as QAM. The extension of SLI from ASK to QAM isstraightforward. Again, in principle, there is no restriction on how theoverlapping subsets are defined. In one embodiment, the nested subsetdesign of 2′-ASK can be generalized to 2^(2L)-QAM. Just as QAM can beviewed as taking the product of two ASK constellations to produce thecomplex QAM constellation, the product of the ASK subsets can be takento produce the subsets of QAM.

FIG. 8 illustrates an embodiment of generalizing ASK to 16-QAM. Each QAMconstellation point shown in FIG. 8 is represented by an ‘X’. The threesubsets for ASK, e.g. as shown in FIG. 4, yield nine subsets for 16-QAMas illustrated by the boxes drawn around different groups ofconstellation points in FIG. 8. Two or more adjacent subsets haveoverlapping constellation points. The middle subset coincides with4-QAM, or QPSK (Quadrature Phase-Shift Keying). The nine centroid valuesfor 16-QAM, which are shown as circles in FIG. 8, are also determinedfrom the respective three centroid values for ASK. ASK can be furthergeneralized to 64-QAM.

The design of overlapping subsets need not be based on the component ASKconstellation. FIG. 9 illustrates an embodiment of the 16-QAMconstellation where each of the subsets is directly determined from theQAM constellation and not derived from ASK. Each QAM constellation pointshown in FIG. 9 is represented by an ‘X’ and the subsets are shown asboxes drawn around different groups of constellation points. Again, atleast two adjacent subsets have overlapping constellation points. Eachof the SLI equalization embodiments described herein, including subsetselection, yield a low complexity alternative to MLSE with goodperformance. SLI-based equalization provides a distinct complexityadvantage over MLSE as the state space grows, either with theconstellation size or with the channel memory.

Storage complexity of MLSE is driven by the number of states, and thecomputational complexity by the number of branches. The states arereferred to next as s and the branches as b. For a conventional MLSEstructure, an estimate of complexity is given by:

q^(M)s+q^(M+1)b  (13)

The complexity of the SLI equalization structures described herein canbe estimated by adding the number of states and the number of branchesfrom the multiple stages. This estimate proves to be over-conservativewhen memory is re-used because the number of states need no be includedin the analysis. Regardless, for N-stage SLI-based equalization, thecomplexity estimate is given by:

$\begin{matrix}{{\sum\limits_{i = 1}^{N}{\left( q^{\prime {\lbrack i\rbrack}} \right)^{M}s}} + {\left( q^{\prime {\lbrack i\rbrack}} \right)^{M + 1}b}} & (14)\end{matrix}$

For 16-QAM, a 2-stage SLI structure may be used based on the ASK-basedoverlapping subsets derived as previously described herein. Such a2-stage SLI structure yields q′^([1])=9 and q′^([2])=4. For memory M=1,the complexity estimate is then given by:

(9s+81b)+(4s+16b)=13s+97b  (15)

In contrast, conventional MLSE requires 16s+256b. As M grows, thecomplexity advantage of SLI-based equalization becomes even moreevident.

For 64-QAM, either a 2-stage or a 3-stage SLI equalization structure maybe used. The 2-stage SLI structure is based on the 8-ASK examplepreviously described herein, with q′^([1])=9 and q′^([2])=16. For M=1,the complexity estimate for the 64-QAM 2-stage SLI structure is givenby:

(9s+81b)+(16s+256b)=25s+337b  (16)

The 3-stage SLI equalization structure uses a 3-level partition for64-QAM, with q′^([1])=9 and q′^([2])=16. The complexity estimate for the64-QAM 3-stage SLI structure is given by:

2(9s+81b)+(4s+16b)=22s+178b  (17)

In contrast, conventional MLSE requires 64s+4096b. The complexityadvantage of SLI is now evident even for M=1, due to the largeconstellation size.

The SLI-based equalization structures described herein tend to work moreefficiently when the self-interference from lagging channel taps isrelatively small. Accordingly, it is useful to combine SLI with apre-filter, whose job is to push the energy of the channel towards theleading tap(s). Doing so increases the total effective channel memory.However, SLI still provides a large complexity advantage over MLSE, evenwith the larger memory. A DFSE structure based on SLI, which isdiscussed later herein, further takes advantage of the pre-filter effectwhile trimming memory size.

FIG. 10 illustrates an embodiment of an SLI equalizer structure 800including a pre-filter 810 and an N-stage SLI equalizer 820 of the kindpreviously described herein. The pre-filter 810 can be designed on thefly, for each realization of the channel response. The effective channelresponse, as determined by a conventional channel estimator 830, is theconvolution of the original channel response and the pre-filter 810, andhas a larger memory than the original, but with more of its energy inthe leading taps, in particular the first tap. A pre-filter designmodule 830 determines the pre-filter coefficients so that the pre-filter810 concentrates the channel energy into the first one or more taps. Thefilter output r′, along with the effective channel tap information, isinput to the N-stage SLI equalizer 820 for performing demodulation andISI suppression as previously described herein. The pre-filter 810 canbe flexibly applied to select stages of the N-stage SLI equalizer 820.That is, some stages, in particular early ones, may be empirically oranalytically determined to be more sensitive to self-interference fromlagging taps, while other stages, in particular later stages, are not.Flexibly applying the pre-filter 810 to particular stages of the SLIequalizer 820 enables more or less complexity and performance to bedesigned into the SLI equalizer structure 800.

As described previously herein, individual SLI equalization stages mayimplement MLSE to perform ISI suppression and demodulation.Alternatively, individual SLI equalization stages can implement DFSE orother types of sequence estimation techniques. DFSE is a very effectiveapproximation to MLSE, offering suitable tradeoff between complexity andperformance.

FIG. 11 illustrates an embodiment where the MLSE component 610 includedin the i-th SLI equalization stage 600 of FIG. 6 is replaced with a DFSEcomponent 900. The DFSE memory M′<M is typically a design parameter. TheDFSE trellis maintained by the DFSE component 900 is the same as theMLSE trellis, with q^(M′) states. Referring to the MLSE branch metricexpression (2), the symbols (ŝ_(k-M), . . . , ŝ_(k-1)) represent theprevious state as well as the present symbol ŝ_(k) in order to computethe branch metric. With DFSE, the previous state represents only the M′most recent symbols (ŝ_(k-M′), . . . , ŝ_(k-1)).

In order to produce the older (M−M′) symbols, tentative symbol decisionsare made, e.g. as explained earlier with regard to MLSE. That is, theDFSE component 900 traces back from the truncated state (ŝ_(k-M′), . . ., ŝ_(k-1)) by following the chain of predecessor states to producetentative decisions, denoted as ( s _(k-M), . . . , s _(k-M′-1)). Assuch, all M symbols are available to compute the branch metric. Sincesome of the tentative decisions ( s _(k-M), . . . , s _(k-M′-1)) may beerroneous, it is desirable to mitigate their contribution to the branchmetric as much as possible. The pre-filter 810 described above issuitable for this purpose.

With regard to the SLI equalization structures described herein, in agiven stage, MLSE can be replaced with DFSE, with or without apre-filter. This way, DFSE can be flexibly chosen for certain stages,but not others. For instance, a certain stage may have a relativelylarge effective constellation, and DFSE would help control thecomplexity of that stage.

With the above range of variations and applications in mind, it shouldbe understood that the present invention is not limited by the foregoingdescription, nor is it limited by the accompanying drawings. Instead,the present invention is limited only by the following claims, and theirlegal equivalents.

What is claimed is:
 1. A method of equalizing inter-symbol interference(ISI) in a received signal corresponding to a transmitted signal carriedover a channel, comprising: grouping points of a constellationassociated with the transmitted signal into a plurality of subsets, atleast two adjacent ones of the subsets having one or more commonconstellation points so that the at least two adjacent subsets overlap;determining a centroid-based value for each of the subsets ofconstellation points; grouping the centroid-based values into one ormore sets for input to an equalizer having a plurality of stages; andequalizing the ISI using the equalizer, each of the stages except for alast one of the stages localizing a search for a final symbol decisionusing the set of centroid-based values input to or selected by the stageas constellation points and the last stage determining the final symboldecision using the subset of constellation points input to or selectedby the last stage.
 2. The method of claim 1, wherein the constellationassociated with the transmitted signal is an effective constellationdetermined for the channel over which the transmitted signal is carried.3. The method of claim 1, wherein the constellation associated with thetransmitted signal corresponds to a constellation used to modulate thetransmitted signal prior to transmission.
 4. The method of claim 1,wherein a first stage of the equalizer localizes the search for thefinal symbol decision by: demodulating the received signal using the setof centroid-based values input to the first stage to generate a sequenceof symbols output by the first stage; generating a re-modulated signalassociated with the first stage as a function of the sequence of symbolsgenerated by the first stage and the channel over which the transmittedsignal is carried; and removing the re-modulated signal associated withthe first stage from the transmitted signal to generate a modifiedsignal for input to the stage immediately following the first stage. 5.The method of claim 4, wherein the equalizer has one or more interveningstages coupled between the first stage and the last stage, each one ofthe intervening stages localizing the search for the final symboldecision by: demodulating a modified version of the received signaloutput by the stage immediately preceding the intervening stage usingthe set of centroid-based values input to or selected by the interveningstage to generate a sequence of symbols output by the intervening stage;generating a re-modulated signal associated with the intervening stageas a function of the channel and the sequence of symbols generated bythe intervening stage; and removing the re-modulated signal associatedwith the intervening stage from the modified version of the transmittedsignal output by the immediately preceding stage to generate a newlymodified version of the transmitted signal for input to the stageimmediately following the intervening stage.
 6. The method of claim 4,wherein the last stage of the equalizer determines the final symboldecision by: demodulating a modified version of the received signaloutput by the stage immediately preceding the last stage using thesubset of constellation points input to or selected by the last stage togenerate a sequence of symbols associated with the last stage; andsumming each of the sequences of symbols generated by the differentstages to determine the final symbol decision.
 7. The method of claim 6,wherein the subset of constellation points input to or selected by thelast stage is the subset having a centroid-based value equal to zero. 8.The method of claim 1, wherein the transmitted signal is modulated witha 2^(L) ASK modulation scheme, the equalizer comprises N of the stages,the set of centroid-based values input to or selected by the ith stagehas centroid-based values of {−2^(N-i), 0, +2^(N-i)} and the subset ofconstellation points input to or selected by the last stage has aconstellation of 2^(L-N+1) ASK, and wherein a leftmost one of thesubsets available for input to or selection by the last stagecorresponds to a centermost one of the subsets shifted by a left one ofthe centroid-based values and a rightmost one of the subsets availablefor input to or selection by the last stage corresponds to thecentermost subset shifted by a right one of the centroid-based values.9. The method of claim 1, wherein the transmitted signal is modulatedwith a QAM scheme, K sets of ASK-based centroids are determined from anASK modulation scheme and M sets of QAM-based centroids are determinedfrom the K sets of ASK centroids for input to the equalizer asconstellation points where M=K².
 10. The method of claim 1, comprisingselecting the subset of constellation points used by the last stage ofthe equalizer for determining the final symbol decision based on amodified sequence of symbols output by the stage immediately precedingthe last stage, the modified sequence of symbols corresponding to one ofthe centroid-based values included in the set of centroid-based valuesinput to or selected by the immediately preceding stage.
 11. The methodof claim 10, comprising determining the final symbol decision based onthe received signal and the subset of constellation points selected forthe last stage of the equalizer.
 12. The method of claim 1, furthercomprising pre-filtering the received signal prior to the receivedsignal being input to the equalizer.
 13. The method of claim 1, whereindetermining a centroid-based value for each of the subsets ofconstellation points comprises determining a centroid for each of thesubsets of constellation points.
 14. A receiver, comprising: aconstellation processing module operable to group points of aconstellation associated with a transmitted signal into a plurality ofsubsets, at least two adjacent ones of the subsets having one or morecommon constellation points so that the at least two adjacent subsetsoverlap, determine a centroid-based value for each of the subsets ofconstellation points and group the centroid-based values into one ormore sets; and a plurality of equalization stages, each of theequalization stages except for a last one of the equalization stagesbeing operable to localize a search for a final symbol decision usingthe set of centroid-based values input to or selected by theequalization stage as constellation points and the last equalizationstage being operable to determine the final symbol decision using thesubset of constellation points input to or selected by the lastequalization stage.
 15. The receiver of claim 14, wherein theconstellation associated with the transmitted signal is an effectiveconstellation determined for a channel over which the transmitted signalis carried.
 16. The receiver of claim 14, wherein the constellationassociated with the transmitted signal corresponds to a constellationused to modulate the transmitted signal prior to transmission.
 17. Thereceiver of claim 14, wherein a first one of the equalization stages isoperable to demodulation a received signal using the set ofcentroid-based values input to the first equalization stage to generatea sequence of symbols output by the first equalization stage, generate are-modulated signal associated with the first equalization stage as afunction of the sequence of symbols generated by the first equalizationstage and a channel over which the transmitted signal is carried, andremove the re-modulated signal associated with the first equalizationstage from the received signal to generate a modified signal for inputto the stage immediately following the first equalization stage.
 18. Thereceiver of claim 17, wherein the receiver comprises one or moreintervening equalization stages coupled between the first and lastequalization stages, each one of the intervening equalization stagesbeing operable to demodulate a modified version of the received signaloutput by the equalization stage immediately preceding the interveningequalization stage using the set of centroid-based values input to orselected by the intervening equalization stage to generate a sequence ofsymbols output by the intervening equalization stage, generate are-modulated signal associated with the intervening equalization stageas a function of the channel and the sequence of symbols generated bythe intervening equalization stage, and remove the re-modulated signalassociated with the intervening equalization stage from the modifiedversion of the received signal output by the immediately precedingequalization stage to generate a newly modified version of the receivedsignal for input to the stage immediately following the interveningstage.
 19. The receiver of claim 17, wherein the last equalization stageis operable to demodulate a modified version of the received signaloutput by the equalization stage immediately preceding the lastequalization stage using the subset of constellation points input to orselected by the last equalization stage to generate a sequence ofsymbols associated with the last equalization stage, and sum each of thesequences of symbols generated by the different equalization stages todetermine the final symbol decision.
 20. The receiver of claim 19,wherein one or more of the equalization stages comprises a maximumlikelihood sequence estimator operable to generate the correspondingsequences of symbols.
 21. The receiver of claim 19, wherein one or moreof the equalization stages comprises a decision-feedback sequenceestimator operable to generate the corresponding sequences of symbols.22. The receiver of claim 19, wherein the subset of constellation pointsinput to or selected by the last equalization stage is the subset havinga centroid-based value equal to zero.
 23. The receiver of claim 14,wherein the transmitted signal is modulated with a 2^(L) ASK modulationscheme, the receiver comprises N of the equalization stages, the set ofcentroid-based values input to or selected by the ith equalization stagehas centroid-based values of {−2^(N-i), 0, +2^(N-i)} and the subset ofconstellation points input to or selected by the last equalization stagehas a constellation of 2^(L-N+1) ASK, and wherein a leftmost one of thesubsets available for input to or selection by the last equalizationstage corresponds to a centermost one of the subsets shifted by a leftone of the centroid-based values and a rightmost one of the subsetsavailable for input to or selection by the last equalization stagecorresponds to the centermost subset shifted by a right one of thecentroid-based values.
 24. The receiver of claim 14, wherein thetransmitted signal is modulated with a QAM scheme and the constellationprocessing module is operable to determine K sets of ASK-based centroidsfrom an ASK modulation scheme and determine M sets of QAM-basedcentroids from the K sets of ASK centroids for input to or selection byeach of the equalization stages except for the last equalization stagewhere M=K².
 25. The receiver of claim 14, wherein the last equalizationstage is operable to select the subset of constellation points used fordetermining the final symbol decision based on a modified sequence ofsymbols output by the equalization stage immediately preceding the lastequalization stage, the modified sequence of symbols corresponding toone of the centroid-based values included in the set of centroid-basedvalues input to or selected by the immediately preceding equalizationstage.
 26. The receiver of claim 25, wherein the last equalization stageis operable to determine the final symbol decision based on a receivedsignal and the subset of constellation points selected by the lastequalization stage.
 27. The receiver of claim 14, further comprising apre-filter operable to pre-filter a received signal prior to thereceived signal being input to a first one of the equalization stages.